Lecture 15: Nonclassically driven atoms (cascaded quantum systems)

Author: J. R. Johansson ([email protected]), http://dml.riken.jp/~rob/

The latest version of this IPython notebook lecture is available at http://github.com/jrjohansson/qutip-lectures.

The other notebooks in this lecture series are indexed at http://jrjohansson.github.com.

In [1]:
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
In [2]:
from scipy import *
In [3]:
from qutip import *

Introduction

In Chapter 12 (Cascaded quantum systems) in Quantum Noise by Gardiner and Zoller (Springer, 3rd edition), a few examples of nonclassically driven atoms are given. In this notebook we solve for the dynamics of those systems using QuTiP.

Two-level atom driven by squeezed light

The master equation for a two-level atom driven by a squeezed light can be written as (Ch. 12.2.2 in Quantum Noise)

$$ \dot\rho = -i[H, \rho] + \kappa\mathcal{D}[a]\rho + \gamma\mathcal{D}[\sigma_-]\rho -\sqrt{\eta\kappa\gamma}\{[\sigma_+, a\rho] + [\rho a^\dagger, \sigma_-]\} $$

where

$$ H = i\frac{1}{2}(E {a^\dagger}^2 - E^* a^2) $$

and

$$ \mathcal{D}[a]\rho = a \rho a^\dagger - \frac{1}{2}\rho a^\dagger a - \frac{1}{2}a^\dagger a\rho $$
$$ \dot\rho = -i[H, \rho] + \kappa\mathcal{D}[a]\rho + \gamma\mathcal{D}[\sigma_-]\rho -\sqrt{\eta\kappa\gamma}\{\sigma_+a\rho - a\rho\sigma_+ + \rho a^\dagger\sigma_- - \sigma_-\rho a^\dagger\} $$
In [4]:
N = 10
gamma = 1
eta = 0.9
In [5]:
def solve(N, gamma, kappa, eta):
    
    E = kappa * 0.25
    
    # create operators
    a = tensor(destroy(N), identity(2))
    sm = tensor(identity(N), destroy(2))

    # Hamiltonian
    H = 0.5j * (E * a.dag() ** 2 - conjugate(E) * a ** 2)

    # master equation superoperators
    L0 = liouvillian(H, [sqrt(kappa) * a, sqrt(gamma) * sm])
    L1 = - sqrt(kappa * gamma * eta) * (
        spre(sm.dag() * a) - spre(a) * spost(sm.dag()) + 
        spost(a.dag() * sm) - spre(sm) * spost(a.dag()))
    
    L = L0 + L1
    
    # steady state
    rhoss = steadystate(L)
    
    # correlation function and spectrum
    taulist = linspace(0, 500, 2500)
    c = correlation_2op_1t(L, rhoss, taulist, [], sm.dag(), sm)
    w, S = spectrum_correlation_fft(taulist, c)
    
    ww = hstack([fliplr(-array([w])).squeeze(), w])
    SS = hstack([fliplr(array([S])).squeeze(), S])

    return rhoss, ww, SS
In [6]:
rhoss2, w2, S2 = solve(N, gamma, 2, eta)
rhoss4, w4, S4 = solve(N, gamma, 4, eta)
rhoss8, w8, S8 = solve(N, gamma, 8, eta)
In [7]:
wigner_fock_distribution(rhoss2.ptrace(0));
wigner_fock_distribution(rhoss4.ptrace(0));
wigner_fock_distribution(rhoss8.ptrace(0));
/usr/local/lib/python3.4/dist-packages/qutip/visualization.py:869: UserWarning: Deprecated: Use plot_wigner_fock_distribution
  warnings.warn("Deprecated: Use plot_wigner_fock_distribution")
In [8]:
fig, ax = plt.subplots()
ax.plot(w2, S2 / S2.max(), label=r'$\kappa = 2$')
ax.plot(w4, S4 / S4.max(), label=r'$\kappa = 4$')
ax.plot(w8, S8 / S8.max(), label=r'$\kappa = 8$')
ax.plot(w8, 0.25/((0.5 * gamma)**2 + w8**2), 'k:', label=r'Lorentian')
ax.legend()
ax.set_ylabel(r'Flouresence spectrum', fontsize=16)
ax.set_xlabel(r'$\omega$', fontsize=18)
ax.set_xlim(-2, 2);

Fig. 12.4 in Quantum Noise.

Two-level atom driven by antibunched light: coherent excitation of the source atom

The master equation given in Sec. 12.3.1 in Quantum Noise, for two coupled atoms where the first atom (source atom) is irradiated with coherent light, and the second atom is irradiated by the antibunched light emitted from the source atom, is:

$$ \dot\rho = -i[H, \rho] + \gamma_1\mathcal{D}[\sigma^-_{1}]\rho + \gamma_2\mathcal{D}[\sigma^-_{2}]\rho -\sqrt{(1-\epsilon_1)(1-\epsilon_2)\gamma_1\gamma_2} ([\sigma_2^+, \sigma_1^-\rho] + [\rho\sigma_1^+, \sigma_2^-]) $$

where

$$ H = -i\sqrt{\epsilon_1\gamma_1}(E\sigma_1^+ - E^*\sigma_1^-) $$
In [9]:
e1 = 0.5
e2 = 0.5

gamma1 = 2
gamma2 = 2

E = 2 / sqrt(e1 * gamma1)
In [10]:
sm1 = tensor(destroy(2), identity(2))
sp1 = sm1.dag()
sm2 = tensor(identity(2), destroy(2))
sp2 = sm2.dag()
In [11]:
H = -1j * sqrt(e1 * gamma1) * (E * sp1 - conjugate(E) * sm1)
In [12]:
L0 = liouvillian(H, [sqrt(gamma1) * sm1, sqrt(gamma2) * sm2])
In [13]:
L1 = - sqrt((1 - e1) * (1 - e2) * gamma1 * gamma2) * \
        (spre(sp2 * sm1) - spre(sm1) * spost(sp2) + 
         spost(sp1 * sm2) - spre(sm2) * spost(sp1))
In [14]:
L = L0 + L1
In [15]:
# steady state
rhoss = steadystate(L)
In [16]:
# correlation function and spectrum
taulist = linspace(0, 4, 250)
In [17]:
G2_11 = correlation_4op_1t(L, rhoss, taulist, [], sp1, sp1, sm1, sm1)
g2_11 = G2_11 / (expect(sp1*sm1, rhoss) * expect(sp1*sm1, rhoss))
In [18]:
G2_22 = correlation_4op_1t(L, rhoss, taulist, [], sp2, sp2, sm2, sm2)
g2_22 = G2_22 / (expect(sp2*sm2, rhoss) * expect(sp2*sm2, rhoss))
In [19]:
G2_12 = correlation_4op_1t(L, rhoss, taulist, [], sp2, sp1, sm1, sm2)
g2_12 = G2_12 / (expect(sp1*sm1, rhoss) * expect(sp2*sm2, rhoss))
In [20]:
G2_21 = correlation_4op_1t(L, rhoss, taulist, [], sp1, sp2, sm2, sm1)
g2_21 = G2_21 / (expect(sp2*sm2, rhoss) * expect(sp1*sm1, rhoss))
In [21]:
fig, ax = plt.subplots()

ax.plot(taulist, g2_11, label=r'$g^{(2)}_{11}(\tau)$')
ax.plot(taulist, g2_22, label=r'$g^{(2)}_{22}(\tau)$')
ax.plot(taulist, g2_12, label=r'$g^{(2)}_{12}(\tau)$')
ax.plot(taulist, g2_21, label=r'$g^{(2)}_{21}(\tau)$')

ax.legend(loc=4)
ax.set_xlabel(r'$\tau$');

Fig. 12.6 in Quantum Noise.

Two-level atom driven by antibunched light: incoherent excitation of the source atom

When the source atom is irradiated with incoherent light, the master equation becomes (Sec. 12.3.2 in Quantum Noise)

$$ \dot\rho = \gamma_1\mathcal{D}[\sigma^-_{1}]\rho + \gamma_2\mathcal{D}[\sigma^-_{2}]\rho + \kappa(\bar{N} + 1)\mathcal{D}[a]\rho + \kappa\bar{N}\mathcal{D}[a^\dagger]\rho -\sqrt{2\kappa\eta_1\gamma_1} ([\sigma_1^+, a\rho] + [\rho a^\dagger, \sigma_1^-]) -\sqrt{\eta_2\gamma1\gamma_2} ([\sigma_2^+, \sigma_1^-\rho] + [\rho\sigma_1^+, \sigma_2^-]) $$
In [22]:
N = 10

e1 = 0.5
e2 = 0.5
ek = 0.5

n_th = 1
kappa = 0.1
gamma1 = 1
gamma2 = 1

E = 0.025

taulist = linspace(0, 5, 250)
In [23]:
a   = tensor(destroy(N), identity(2), identity(2))
sm1 = tensor(identity(N), destroy(2), identity(2))
sp1 = sm1.dag()
sm2 = tensor(identity(N), identity(2), destroy(2))
sp2 = sm2.dag()
In [24]:
def solve(ek, e1, e2, gamma1, gamma2, kappa, n_th, E):
    
    eta1 = (1 - ek) * e1
    eta2 = (1 - e1) * (1 - e2)
    
    H = 1j * E * (a - a.dag())
    
    L0 = liouvillian(H, [sqrt(kappa * (1 + n_th)) * a, sqrt(kappa * n_th) * a.dag(),
                     sqrt(gamma1) * sm1, sqrt(gamma2) * sm2])
    
    L1 = - sqrt(2 * kappa * eta1 * gamma1) * \
            (spre(sp1 * a) - spre(a) * spost(sp1) + 
             spost(a.dag() * sm1) - spre(sm1) * spost(a.dag())) + \
         - sqrt(eta2 * gamma1 * gamma2) * \
            (spre(sp2 * sm1) - spre(sm1) * spost(sp2) + 
             spost(sp1 * sm2) - spre(sm2) * spost(sp1))
    
    L = L0 + L1
    
    rhoss = steadystate(L)
    
    G2_11 = correlation_4op_1t(L, rhoss, taulist, [], sp1, sp1, sm1, sm1)
    g2_11 = G2_11 / (expect(sp1*sm1, rhoss) * expect(sp1*sm1, rhoss))

    G2_22 = correlation_4op_1t(L, rhoss, taulist, [], sp2, sp2, sm2, sm2)
    g2_22 = G2_22 / (expect(sp2*sm2, rhoss) * expect(sp2*sm2, rhoss))

    G2_12 = correlation_4op_1t(L, rhoss, taulist, [], sp2, sp1, sm1, sm2)
    g2_12 = G2_12 / (expect(sp1*sm1, rhoss) * expect(sp2*sm2, rhoss))
    
    G2_21 = correlation_4op_1t(L, rhoss, taulist, [], sp1, sp2, sm2, sm1)
    g2_21 = G2_21 / (expect(sp2*sm2, rhoss) * expect(sp1*sm1, rhoss))
    
    return rhoss, g2_11, g2_12, g2_21, g2_22
In [25]:
# thermal
rhoss_t, g2_11_t, g2_12_t, g2_21_t, g2_22_t = solve(ek, e1, e2, gamma1, gamma2, 
                                                    kappa, n_th, 0.0)
In [26]:
# visualize the cavity state
wigner_fock_distribution(rhoss_t.ptrace(0));
In [27]:
fig, ax = plt.subplots(figsize=(8,4))

ax.plot(taulist, g2_11_t, label=r'$g^{(2)}_{11}(\tau)$')
ax.plot(taulist, g2_22_t, label=r'$g^{(2)}_{22}(\tau)$')
ax.plot(taulist, g2_12_t, label=r'$g^{(2)}_{12}(\tau)$')
ax.plot(taulist, g2_21_t, label=r'$g^{(2)}_{21}(\tau)$')

ax.legend(loc=4)
ax.set_xlabel(r'$\tau$', fontsize=16);

Similar to Fig. 12.8 in Quantum Noise, although not exactly because of different parameters.

Versions

In [28]:
from qutip.ipynbtools import version_table; version_table()
Out[28]:
SoftwareVersion
IPython2.0.0
OSposix [linux]
Numpy1.8.1
Cython0.20.1post0
QuTiP3.0.0.dev-5a88aa8
SciPy0.13.3
matplotlib1.3.1
Python3.4.1 (default, Jun 9 2014, 17:34:49) [GCC 4.8.3]
Thu Jun 26 14:13:04 2014 JST